Is this statement true?

Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.

If $\varphi$ is polynomially growing and of infinite order, then the semidirect product $\mathbb{F}\rtimes_\Phi\mathbb{Z}$ is acylindrically hyperbolic but not relatively hyperbolic.

If there is anything unclear about notation, please tell me. This statement is closely related to Problem 8.2 in A. Minasyan and D. Osin. Acylindrical hyperbolicity of groups acting on trees. math. Annalen, 362:1055–1105, 2015 (arXiv link). This problem asks which mapping tori of injective endomorphisms of finitely generated free groups are acylindrically hyperbolic.

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    $\begingroup$ When will you stop daily minor unnecessary edits to your posts? You've just edited 3 posts with, changing ":" into "?" in one, changing "such a" to "the" in another, and moving the introductory sentence here into the grey rectangle. Each time the previous wording was correct. You already did such unnecessary edits 2 days ago and before to the same posts (including one edit adding a space before ?, followed later by the reverse edit). The only goal is to appear in the front list? You're not the first one, see meta.mathoverflow.net/questions/2749/… $\endgroup$ – YCor Mar 17 '18 at 6:44

This was proved by Jack Button and Robert Kropholler: arXiv link, see p.27. (Added: But there's a caveat; see the second update below.)

J.O. Button, R. Kropholler Nonhyperbolic free-by-cyclic and one-relator groups. New York J. Math. 22 (2016), 755-774. Journal link (unrestricted access).


As YCor notes in the comments, the result that the group is not relatively hyperbolic does not appear in the published version of the Button--Kropholler paper. However, Mark Hagen has pointed out to me in a personal communication that this fact can be deduced from known results. Macura showed that these groups have polynomial divergence, while a folklore theorem, written up by Sisto (Theorem 6.13 of this arXiv link), asserts that non-elementary relatively hyperbolic groups have exponential divergence. So the fact follows.

Second update:

A personal communication from Jack Button has provided two more useful pieces of information.

First, the proof of non-relative-hyperbolicity in the arXiv version is correct; it was only removed from the published version for reasons of space, and because the result was already attributed to Macura.

Second, and more importantly, the published Button--Kropholler paper actually only proves that the free-by-cyclic groups under discussion are virtually acylindrically hyperbolic. In fact, it's still open whether virtually acylindrically hyperbolic groups are acylindrically hyperbolic, in general.

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    $\begingroup$ I've added the published reference, see esp. p772 (=p18 of pdf). Compared to the preprint, the relative hyperbolicity part (theorem 4.4 in the arxiv preprint) has disappeared, and only remains the result that the groups considered by the OP are acylindrically hyperbolic. Henry, do you know more about the picture? $\endgroup$ – YCor Mar 15 '18 at 9:00
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    $\begingroup$ I'm not sure, but am investigating... It seems this fact can also be deduced from other work in the literature. I'll try to provide a reference when my investigations are concluded. $\endgroup$ – HJRW Mar 15 '18 at 11:13
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    $\begingroup$ Thanks for the additional details! Macura's ref: Natasa Macura, Detour functions and quasi‐isometries. Quarterly J. of Math., 53(2), 2002, 207–239 $\endgroup$ – YCor Mar 15 '18 at 23:00

In addition to Henry's answer, I would like to mention that the acylindrical hyperbolicity of free-by-cyclic groups is fully characterised in the recent preprint Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups, allowing us to remove the word virtually in Button and Kropholler's statement. Actually, we have the more general proposition:

Theorem: Fix a group $H$ and a morphism $\varphi : H \to \mathrm{Aut}(\mathbb{F}_n)$ where $n \geq 2$. The semidirect product $\mathbb{F}_n \rtimes_\varphi H$ is acylindrically hyperbolic if and only if $$\mathrm{ker} \left( H \overset{\varphi}{\to} \mathrm{Aut}(G) \to \mathrm{Out}(G) \right)$$ is a finite subgroup of $H$.

In the particular case where $H= \mathbb{Z}$, we find that $\mathbb{F}_n \rtimes_\Phi \mathbb{Z}$ is acylindrically hyperbolic if and only if $\Phi$ has infinite order in $\mathrm{Out}(\mathbb{F}_n)$.

Such a behaviour seems to be quite common. The same statements hold for infinitely-ended groups, non-elementary hyperbolic groups, most non-elementary relatively hyperbolic groups (including toral relatively hyperbolic groups), irreducible right-angled Artin/Coxeter groups (and more generally, most graph products of groups; see this preprint).


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